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Applying Universal Algebra to Lambda Calculus
, 2007
"... The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to se ..."
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The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λtheories ( = equational extensions of untyped λcalculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art about the longstanding open questions concerning the representability of λtheories as theories of models, and 26 open problems. On the other side, against the common belief, we show that lambda calculus and combinatory logic satisfy interesting algebraic properties. In fact the Stone representation theorem for Boolean algebras can be generalized to combinatory algebras and λabstraction algebras. In every combinatory and λabstraction algebra there is a Boolean algebra of central elements (playing the role of idempotent elements in rings). Central elements are used to represent any combinatory and λabstraction algebra as a weak Boolean product of directly indecomposable algebras (i.e., algebras which cannot be decomposed as the Cartesian product of two other nontrivial algebras). Central elements are also used to provide applications of the representation theorem to lambda calculus. We show that the indecomposable semantics (i.e., the semantics of lambda calculus given in terms of models of lambda calculus, which are directly indecomposable as combinatory algebras) includes the continuous, stable and strongly stable semantics, and the term models of all semisensible λtheories. In one of the main results of the paper we show that the indecomposable semantics is equationally incomplete, and this incompleteness is as wide as possible.
Resource combinatory algebras
"... Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down fou ..."
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Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λcalculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambdaalgebras and resource lambdaabstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λcalculus. We also show that the ideal completion of a resource combinatory (resp. lambda, lambdaabstraction) algebra induces a “classical ” combinatory (resp. lambda, lambdaabstraction) algebra, and that any model of the classical λcalculus raising from a resource lambdaalgebra determines a λtheory which equates all terms having the same Böhm tree. 1
Nominal Algebra and the HSP Theorem
"... Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. ..."
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Nominal algebra is a logic of equality developed to reason algebraically in the presence of binding. In previous work it has been shown how nominal algebra can be used to specify and reason algebraically about systems with binding, such as firstorder logic, the lambdacalculus, or process calculi. Nominal algebra has a semantics in nominal sets (sets with a finitelysupported permutation action); previous work proved soundness and completeness. The HSP theorem characterises the class of models of an algebraic theory as a class closed under homomorphic images, subalgebras, and products, and is a fundamental result of universal algebra. It is not obvious that nominal algebra should satisfy the HSP theorem: nominal algebra axioms are subject to socalled freshness conditions which give them some flavour of implication; nominal sets have significantly richer structure than the sets semantics traditionally used in universal algebra. The usual method of proof for the HSP theorem does not obviously transfer to the nominal algebra setting. In this paper we give the constructions which show that, after all, a ‘nominal ’ version of the HSP theorem holds for nominal algebra; it corresponds to closure under homomorphic images, subalgebras, products, and an atomsabstraction construction specific to nominalstyle semantics. Keywords: universal algebra, equational logic, nominal algebra, HSP or Birkhoff’s theorem, nominal sets, nominal terms 1
The Visser topology of lambda calculus
"... A longstanding open problem in lambda calculus is whether there exists a nonsyntactical model of the untyped lambda calculus whose theory is exactly the least λtheory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a m ..."
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A longstanding open problem in lambda calculus is whether there exists a nonsyntactical model of the untyped lambda calculus whose theory is exactly the least λtheory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a model can be recursively enumerable (r.e. for brevity). We introduce the notion of an effective model of lambda calculus and prove the following results: (i) The equational theory of an effective model cannot be λβ, λβη; (ii) The order theory of an effective model cannot be r.e.; (iii) No effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove the following, where “graph theory ” is a shortcut for “theory of a graph model”: (iv) There exists a minimum order graph theory (for equational graph theories this was proved in [9, 10]). (v) The minimum equational/order graph theory is the theory of an effective graph model. (vi) No order graph theory can be r.e. (vii) Every equational/order graph theory is the theory of a graph model having a countable web. This last result proves that the class of graph models enjoys a kind of (downwards) LöwenheimSkolem theorem, and it answers positively Question 3 in [4, Section 6.3] for the class of graph models.
From lambda calculus to universal algebra and back
 33rd International Symposium on Mathematical Foundations of Computer Science, LNCS
, 2008
"... We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λtheories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest ..."
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We generalize to universal algebra concepts originating from lambda calculus and programming in order first to prove a new result on the lattice of λtheories, and second a general theorem of pure universal algebra which can be seen as a meta version of the Stone Representation Theorem. The interest of a systematic study of the lattice λT of λtheories grows out of several open problems on lambda calculus. For example, the failure of certain lattice identities in λT would imply that the problem of the orderincompleteness of lambda calculus raised by Selinger has a negative answer. In this paper we introduce the class of Church algebras (which includes all Boolean algebras, combinatory algebras, rings with unit and the term algebras of all λtheories) to model the ifthenelse instruction of programming and to extend some properties of Boolean algebras to general universal algebras. The interest of Church algebras is that each has a Boolean algebra of central elements, which play the role of the idempotent elements in rings. Central elements are the key tool to represent any Church algebra as a weak Boolean product of directly indecomposable Church algebras and to prove the meta representation theorem mentioned above. We generalize the notion of easy λterm and prove that any Church algebra with an “easy set ” of cardinality n admits (at the top) a lattice interval of congruences isomorphic to the free Boolean algebra with n generators. This theorem has the following consequence for λT: for every recursively enumerable λtheory φ and each n, there is a λtheory φn ≥ φ such that {ψ: ψ ≥ φn} “is ” the Boolean lattice with 2 n elements. 1.
Effective λmodels versus recursively enumerable λtheories
"... A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively e ..."
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A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λcalculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem.
Algebra and Topology in Lambda Calculus The
"... untyped lambda calculus was introduced around 1930 by Church [11] as part of an investigation in the formal foundations of mathematics and logic. Although lambda calculus is a very basic language, it is sufficient to express all the computable functions. The process of application and evaluation ref ..."
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untyped lambda calculus was introduced around 1930 by Church [11] as part of an investigation in the formal foundations of mathematics and logic. Although lambda calculus is a very basic language, it is sufficient to express all the computable functions. The process of application and evaluation reflects the computational behavior of many modern functional programming languages, which explains the interest in the lambda calculus among computer scientists. The lambda calculus, although its axioms are all in the form of equations, is not a true equational theory since the variablebinding properties of lambda abstraction prevent variables in lambda calculus from operating as real algebraic variables. Consequently the general methods that have been developed in universal algebra, for defining the semantics of an arbitrary algebraic theory for instance, are not directly applicable. There have been several attempts to reformulate the lambda calculus as a purely algebraic theory. The earliest, and best known, algebraic models are the combinatory algebras of Curry and Schönfinkel (see [12]). Combinatory algebras have a simple purely equational characterization and were used to provide an intrinsic firstorder, but not equational, characterization of the models of lambda calculus, as a special class of combinatory algebras called λmodels [1, Def. 5.2.7]. Lambda theories are equational extensions of the untyped lambda calculus closed under derivation. They arise by syntactical or semantic considerations. Indeed, a λtheory may correspond to a possible operational semantics of lambda calculus, as well as it may be induced by a model of lambda calculus through the kernel congruence relation of the interpretation function. The set of lambda theories is naturally equipped with a structure of complete lattice (see [1, Chapter 4]), where the meet of a family of lambda theories is their intersection, and the join is the least lambda theory containing their union. The bottom element of this lattice is the minimal lambda theory λβ, while the top element is the inconsistent lambda theory, hereafter denoted by ∇. Although researchers have mainly focused their in
Effective λmodels versus recursively enumerable λtheories
"... Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be rec ..."
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Abstract. A longstanding open problem is whether there exists a nonsyntactical model of the untyped λcalculus whose theory is exactly the least λtheory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λcalculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λcalculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott’s semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards LöwenheimSkolem theorem. key words: λcalculus, effective λmodels, effectively given domains, recursively enumerable λtheories, graph models, LöwenheimSkolem theorem. Contents 1